Divergence and Curl Calculator
Divergence: -
Curl: -
Understanding Vector Calculus
What are Divergence and Curl?
Divergence and curl are fundamental operations in vector calculus that describe the behavior of vector fields.
Divergence: ∇·F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z
Curl: ∇×F = (∂F₃/∂y - ∂F₂/∂z)î + (∂F₁/∂z - ∂F₃/∂x)ĵ + (∂F₂/∂x - ∂F₁/∂y)k̂
where:
- F = F₁î + F₂ĵ + F₃k̂ is the vector field
- ∂/∂x, ∂/∂y, ∂/∂z are partial derivatives
- î, ĵ, k̂ are unit vectors
Physical Interpretations
- Divergence: Measures flux density (source/sink strength)
- Positive divergence: Source (outward flow)
- Negative divergence: Sink (inward flow)
- Zero divergence: Incompressible flow
- Curl: Measures rotation tendency
- Curl magnitude: Rotation strength
- Curl direction: Rotation axis
- Zero curl: Irrotational/conservative field
Advanced Concepts
Gauss's Theorem
∭(∇·F)dV = ∯F·dS
Stokes' Theorem
∯(∇×F)·dS = ∮F·dr
Green's Theorem
∮F·dr = ∬(∂Q/∂x - ∂P/∂y)dA
Helmholtz Decomposition
F = ∇φ + ∇×A
Applications
- Fluid Dynamics: Flow analysis
- Electromagnetics: Field theory
- Quantum Mechanics: Operators
- Elasticity: Strain analysis
- Aerodynamics: Air flow
- Plasma Physics: Field evolution
- Geophysics: Field modeling
- Weather Forecasting: Wind fields